Quantum Cluster Theorynon-local corrections to DMFMark JarrellUniversity of Cincinnati

● DCA● Cluster Solvers● Convergence● Outlook

Collaborators and References

● K. Aryanpour● J. Deisz● O. Gonzalez● J. Hague● M. Hettler● C. Huscroft● H.R. Krishnamurthy● A. Macridin● Th. Maier● Th. Pruschke● Th. Schulthess● A.N. Tavilderzahdeh● F.C. Zhang

● Papers and talks (DCA):� www.physics.uc.edu/~jarrell/� www.physics.uc.edu/~jarrell/TALKS/� xxx.lanl.gov

● Figures:� www.lps.u-

psud.fr/Activites/ThemeA.asp● Further reading and Citations

� CDMF Kotliar et al., PRL 2001� MCPA F. Ducastelle, J. Phys. C. 7,

1795 (1974).

Local Approximations

Field

Mean

D=1 D=2 D=3 D=∞

CPADMF

Curie-WeissMigdal-Eliash.

1/D corrections?PvD 1995

The central site has 2D nearest neighbors

...

...

Two Causal Cluster ApproachesDynamical Cluster Approximation Cellular Dynamical Mean Field

Molecular CPA

Effective medium

Cluster

Effective medium

Cluster

L

�x

X

- k= 2Æ/L

�k

First Brillouin zone

K

Ducastelle 74 Kotliar 01

DCA Mapping to Cluster: Coarse Graining

kx

ky

Kx

Ky

M k = Kk

Kk1

k3

k2

- = N º k1� k2 ,k3

Ncº M k1 � M k2 ,M k3

DCA vs. DMFA

k1k3

k5

k2- =1

G k �G r= 0

r=0 r=0

- = N cº M k 1 � M k 2 , M k 3

k1k3

k5

k2G k �G K

Nc=1 DMFA

Nc >1 DCA

K

K' � Q

K� Q

K'

Mueller Hartmann (89)Metzner Vollhardt (89)

V k �V r= 0

V k �V K

V

G

k4k6

k6

k4 Q

Dynamical Cluster Approximation

´ G k ,V k H´ G K ,V k

¬ G k H¬ G,V

¶ = ´ � Tr ² G �Trln �Gº ¶º G

= 0�² G k H² G,V

mapping from the cluster back to the lattice

DCA Algorithm

ClusterSolver

G K

² K = 1G0 K

� 1G K

1G0 K

= ² K � 1G K

G0 K

G G k

Dynamical Cluster Approimation

Effective medium

Cluster

●fully causal●maintains lattice point group symmetries●maintains translational invariance●systematic (DMFA → Nc=1)●converges quickly Γ∝ 1/L2

¬

Cluster Solvers

ClusterSolver

1/G 0 K = ² K � 1/G K

G G k

² K = 1 /G0 K � 1/G K

Quantum Monte CarloFLEX

Non-Crossing ApproximationExact Enumeration

Average over Disorder

Quantum Monte Carlo Cluster Solver

G0QMC ClusterSolver on oneprocessor

G

QMCtimewarmup sample

QMC ClusterSolver on oneprocessor

QMC ClusterSolver on oneprocessor

QMC ClusterSolver on oneprocessor

G0

G

G

G

warmup sampleQMCtime

Serial

Perfectly Parallel

Hybrid Parallel QMC

QMC ClusterSolver on manyprocessors

QMC ClusterSolver on manyprocessors

G0

G

G

warmup sampleQMCtime

G GGG

G GGG

G GGG

G GGG

Perfectly parallel array of cpu's

GHybrid parallel array of cpu's

GG

GOpenMP

orPBLAS

Sign Problem

Finite-Size Simulations (FSS): White (1989)

Phase Diagram for 2DHubbard

FLEX as a Cluster Solver (2D Hubbard)See poster by Karan Aryanpour

Compare Cluster Approaches

Effective medium

Cluster

Effective medium

Cluster¬ ¬

¬� 1L2

L

¬�2D LD�1

LD = 2DL

●maintains point group symmetries●fully causal●violates translational invariance●converges slowly with corrections

●maintains point group symmetries●fully causal●maintains translational invariance●converges quickly with corrections

MCPA/CMDFDCA

USimplified Hubbard Model

Compare Cluster Approaches

UtÜ= t tÝ=0Simplified Hubbard Model

Compare Cluster Approaches

Conclusion•DCA: systematic non-local corrections to the DMFA

•Preserves translational and point group symmetries•Converges quickly (correction O(1/L2 ))•Converges quickly even in 1D.

•Many cluster solvers may be used.•QMC: very mild minus sign problem

•DCA complementary to FSS.•See http://www.physics.uc.edu/~jarrell for more info.

Outlook•MFT for the cuprates (lanl.gov)•LDA+DCA (with Th. Schulthess, ORNL).•DCA for nanotubes (lanl.gov shortly).•QMC+MEM codes available for collaboration.•GPL codes within 2 years (some sooner).

MCA Mapping to Cluster: Molecules

�x

X

x

Lx =

�x � X

Correlations within themolecules are treatedexplicitly; while thosebetween molecules areignored

G X 1 , X 2 ,�x Translational invariance

is violated

MCA vs. DMFA

x1 x2G x 1 , x 2 � G

�x = 0

X= 0 X= 0

G x 1 , x 2 � G X 1 , X 2 ,�x = 0

Nc=1 DMFA

Nc >1 MCA

Molecule Nc >1

Molecule Nc=1

x1 x2

X1 X2

Cellular Dynamical Mean FieldMolecular CPA

´ G x 1 � x 2 H ´ G X 1 , X 2 ,�x = 0

¶ = ´ � Tr ² G �Trln �Gº ¶º G

= 0�² x1� x2 H ² X 1 , X 2 ,�x= 0 º �

x1 ,�

x2

MCA Algorithm

ClusterSolver

G

² =G0�1�G�1G0

�1= ² �G�1

G0

G G

G0 ,² ,G... Are matrices in the cluster coordinates